Good morning, good afternoon, good evening, wherever you on the world, and of course, very warm. Welcome to this Felix live refresher session on option weeks.

My name is Thomas Krausen. I'm head of financial products here at Financial Edge, and I have the honor to deliver the session to you.

Today and just maybe a little bit of background about myself, I started my career in fixed income, mostly trading interest rates and also, uh, foreign exchange and cash and derivatives.

But I also had the opportunity towards the end of my practitioner career to work in a cross asset mandate, which then gave me a good amount of insight into the equity and credit markets, as well.

So let's have a quick look at the agenda today.

As I said, it's gonna be a refresher on option Greeks and this in general means it's going to be relatively fast-paced, and we also assume a good knowledge of options and their general mechanics here because what we wanna do in this session particularly is start with a quick recap on the main option price drivers and then lead this into the concept of sensitivities.

And then with that we're gonna discuss Delta, gamma, CTA, and Vega in a little bit more detail.

But before we start, a couple of, general reminders here. First of all, you can access the course materials.

Now a link has been shared in the chat.

You also find a link in the resource section of the zoom window.

And then finally, you can also access these materials via the Felix Live website.

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And the pro tip, if you wish, if you have any follow up questions you want to ask.

The feedback form is also a great way of asking this.

If you didn't get a chance to ask a question or you can think of something later on, that's a good way. And of course, if there are any additional market related topics that you would like to see covered in this format, then the feedback form is a great way of letting us know.

But I think that should be it in terms of general introductions and reminders.

So without further ado, let's get started.

And as I said, we're going to start by just reminding us of the main option price drivers and how option premium and those price drivers are actually linked together.

And when you learned about options and the mechanics and you discuss the option premium, I'm reasonably convinced that you would've seen a slide At least very similar to this one because what this slide does, it shows the price drivers of the option premium.

That's a spot price, the strike, the time to expiry volatility rates, asset yields like for example, dividends in in case of equity, et cetera. And what we're going, or what we're doing here on this slide is actually that we're analyzing the impact of an increase in each of the factors that is shown in the left hand column here on the price of calls and on the price of puts.

So let's go through this very briefly on the, or for the example of calls i.e. we're looking at the middle column here.

And what we're going to do is we're approaching this by generically looking at intrinsic value and time value.

Because in general what we can say for the option premium is that it's a sum of these two components, right? So the option premium is a sum of intrinsic value, and time value.

So with that in mind, let's go through the different factors and see how they affect the option premium.

And what we're always looking at is an increase in the variable.

So first line item if you wish, we're looking here at what happens to the price of an option if the spot price of the underlying actually goes up.

Now what that does it, it increases the forward price as well, at least all other factors remaining unchanged.

And that in turn will increase the intrinsic value of the option.

Um, and that is then of course through the formula that we have John here at the top of the slide gonna lead to an increase in the option premium.

Now the next factor, the strike price of course usually doesn't change over the life of the option, but what we're assuming here is that we have two call options.

All other factors are identical.

The only difference between those two is that one has a higher strike than the other.

And I hope that it's intuitive that in general we would prefer the call option with the lower strike.

So the higher the strike, the lower the premium of a call than will be. And I think this is quite intuitive because the lower the strike is the higher as your probability that you will be able to exercise this option.

And if you actually exercise the option, that also means you will have a higher potential payoff in this case.

Now, of course, you could argue with the intrinsic value here in a way as well.

So the intrinsic value is the difference between the forward price and the strike price more, you know, exactly we should say it's a present value.

But let's simplify this here.

So the higher the strike often option is the lower the intrinsic value of the option will be, but technically this only works when we're looking at in the money, uh, forward strikers of options.

Anyway, let's have a look at the next factor that brings us to time to expiring.

Now this is clearly linked and the name suggests is to the time value component of the option premium.

And my way to think about the time value of an option is really that it can be best described as sort of like the compensations that the option seller receives for taking the uncertainty about whether or not an option will be exercised.

And this uncertainty clearly increases as time to expiry increases, right? So what that means is if we have two options, everything else is again identical, but they have different times to expiry, then they should not have the same price.

The option with the longer time to expiry bear greater uncertainty for the option seller and therefore should have a higher premium.

And that also kind of links nicely into the, next factor and that is volatility.

And volatility also impacts the time value.

If the volatility of an underlying increases, for example, the uncertainty around whether or not the option will be exercised, it does increase as well.

And so then does the time value with it and then all else equal the option premium will increase as well.

And then the last two factors, interest rates and asset yields, will impact the intrinsic value predominantly again, because what we are basically looking at is, or others cost of carry components that then drive the difference between spot and forward price.

So if interest rates for example, increase the forward price of the asset increases and that will then increase the intrinsic value and therefore the option premium if asset yields i.e. dividends for example increase, this will lower the forward price, therefore the intrinsic value will decrease and the option premium as well.

So this completes the look at the call.

Now let's very, very quickly look to the column on the right hand side where we discuss the same for puts, but I think it's actually quite straightforward to derive how the option premium should be affected by changing all the factors and by just, you know, deriving it straight from the call option. Because in terms of directional risk put options are the exact opposite of call options. The owner of a call benefits from a rising underlying price, the owner of a per benefits from a decline in the underlying price.

So all those factors that impact the intrinsic value and those were spot strike interest rates and asset yield.

In the case of the call, we discussed it in more detail, have the opposite impact on the price of a put, then they have on the price of a column.

You can see if you, for example, look at the impact here, on put price and call price, that's the opposite.

Same for strike, same for rates, dividend yields, that is the exact opposite.

But all those factors that impact the time value, which is not about direction, but the actual uncertainty of the option to be exercised, they impact calls and puts in the exact same way.

So the exposure if you wish to time to expiry and volatility is the same at least directionally for calls and puts.

Alright? Now I think, and that's why I decided to start here, this conceptual understanding is, is really, really important and useful to have and that's why I wanted to make sure we refresh that.

In practice though, what we need is much more detailed and also quantitative information because if you trade options, you usually won't just have a single option position in your portfolio, but you know, quite a few hundreds if not thousands of different positions.

And it makes sense to look at this portfolio or the risk of this portfolio in an aggregated way because some of the risks that the individual option positions will create might at least partially offset each other.

Right? Now think back about what we saw on the previous slide and imagine that you're a market maker in single stock options.

Now also assume you just bought a lot of calls from one client and you bought a lot of puts from another client and both options are on the same underline.

So let's assume we don't want to have directional exposure, i.e. exposure to a change in spot price and we go into details of hedging later, but we know already from, you know, observations on slide one that the long call basically gives you a long exposure to the underline.

And to get rid of that we would have to sell some stock, to hedge this.

And we also know that the long put position gives us a short position in the underlying.

So we would have to buy stocks to cover this and it's easy to see that it makes sense to look at both positions together to figure out the net risk I what's the overall exposure to the spot price before we start hedging anything because every hatch comes with costs and we wanna reduce those transaction costs.

And of course you might say, hey, that's not option specific.

And you're absolutely right.

We would do the exact same approach for a bond book for a portfolio of interest rate swaps, et cetera, et cetera.

So to bring that back into option space, the general approach to managing risk in context of an option portfolio would be first to decompose all single option positions into their sensitivities.

i.e. think about what is my exposure to sport two vol, two interest rates, et cetera for each individual position.

And then in the second step, we are now going to aggregate those sensitivities, those exposures on a portfolio level and at that stage at which we will see the partial offsetting i.e. the long position from the call and the short position from the put will cancel each other out, maybe not perfectly, but at least there will be a reduction.

And then once we know the combined risk positions, we can then efficiently modify those if we want to, right? Because basically what we wanna make sure is that the risk that we're having on our portfolio is in line with our market views, our limits, our risk appetite, et cetera, et cetera.

And if we realize that the current status quo, the risk in our portfolio is not where we would like it to be, then we will have to make the relevant modifications and that could be done by overlay strategies.

Alright? So that's a general approach.

That approach however requires the quantification of risk, right? Because these qualitative statements that we've made on the first slide, they are not really helpful because we cannot really add them up.

Yes, I know I'm long the underlying through the long call and I'm sure the underlying through the long put, but how long, how short am I actually, on a net basis? That's something we need to calculate, right? And this is exactly where then those sensitivity ratios come in.

And in option world, those are numbers that basically show us how sensitive the option premium is to a change in those price drivers that we have already identified.

In other words, they measure how much, for example, the option premium changes when the spot price increases, when interest rates go up, uh, is hedge.

And now in the option world, many of those sensitivities that we're looking at on a regular basis have letters of the Greek alphabet assigned to them.

And that's why we generally call those sensitivities the Greeks.

So instead of saying then the sensitivity of the option premium to a change in spot price, we say Delta right theta measures the impact of time to expiry shortening vigor or sometimes also referred to as kappa measures the impact on the option premium caused by a change in volatility.

Row row two quantify the impact of a change in interest rates, respective asset yields and so on.

Now what you notice here is that obviously for the strike price, there's no sensitivity or no Greek letter assigned and that is relatively clear I think as to why, because as mentioned, the strike price usually doesn't change.

So it's not really a variable, that needs to be calculated in terms of sensitivities.

And when you look at this list of Greek, at least mostly Greek letters here on the, on the on in the sensitivity column, you might feel that, you know, you've heard of other Greeks and you are missing them on this list, right? And you're absolutely right, there are more Greeks usually they are what we call higher order Greeks, Vana Voma and the likes.

So they are a little bit different and we wanted to, in order to be able to squeeze this session into one hour format to focus on you know, mostly first order Greeks, but we're gonna have a look at one higher order Greek more specifically as second order Greek later.

But what we want to do is we wanna start with the first, uh, Greek letter that most option traders would usually address first.

And that is the option Delta.

Now first of all, let's think about, or let's remind you of the definition of option Delta.

Now option Delta generally indicates the sensitivity of the option premium to a change in the underlying price.

In other words, if the spot goes up by, for example, $1, how much does the price of the option or the option premium check? Now Delta in many cases is given to us as a percentage value.

So a value between zero and one or between zero and 100%, which is essentially the same.

But what does Delta value of 50% actually mean or 0.5 for that matter? Well it means that if the price of the underlying increases by $1, the option premium will change by 50% of that.

So basically by 50 cents.

And this is then also why the option Delta is often referred to as a participation ratio, even a likeness factor, it shows us how much like the underlying does the option premium behave.

Alright? Now in case you all you've been given is an absolute number, so there's no indication if Delta is positive or negative, you might have to figure out if Delta should be positive or negative by looking at your option position.

And so the question you have to answer them basically is if spot and the value of the option position move in the same direction or if they move in opposite directions, if they move in the same direction, that means your Delta is positive.

If not Delta is negative.

Let's go through this in case or for the case of a long call position.

Okay? Now as discussed on the first slide, in case of a call and we are long the call, that means that spot price of the underlying and the option premium move in the same direction.

Remember we said if spot price increase, the premium of a call increases. So now we're looking at increase in spot that leads to an increase in the option premium leads in the same direction, the value change of the option goes up as well as positive as well.

That means we have positive Delta on long call options, which can also sometimes be referred to as we are long Delta, but it's not only a long call position but also a short put position that has a positive Delta because in case of a short put, the absolute value of the option premium decreases as the spot goes up.

But we're short the put, so we are the option rider and this then basically means it's beneficial for us.

So an increase in spot basically leads to a positive p and l change for the seller of a put.

And that's why long call and short puts both have a positive or long Delta assigned to them.

And then of course if you know the Delta for a long call, in short put the Delta for long puts and short calls is just the exact opposite.

So now if we go back to our early example of risk aggregation, it becomes perfectly clear how the portfolio approach is beneficial because we said that we were long calls and long puts. Now we're combining this with our knowledge on long and short Delta.

And what we can say is that the long call gives us a long Delta position, whereas a long put gives us a short Delta position.

And so both positions taken together will at least partially of offset each other.

And that's then obviously the increased efficiency that we can take advantage of in the hedging context.

And that leads us nicely to another critical function of option Delta.

It does tell us the hedge ratio for an option and what this means is it tells us the number of units in the underlying asset that we have to buy or sell to hedge this directional risk, of the position.

And this type of transaction is exactly what we are referring to when we talk about Delta hedging.

And the main idea behind a Delta hedge is actually to reduce the portfolio Delta by, for example, trading the underlying directly.

And this delta hedging is pretty much standard market practice amongst many type of market participants, like for example, market makers as it really quickly and also relatively easily eliminates a significant source, of market risk.

But I suggest we have a look at a concrete example just to clarify this.

And the assumption that we're gonna use is that we have bought a hundred calls.

So we're long a hundred calls and the Delta of these calls is 0.5.

What this means, if we just remember the definition of Delta, it sort of tells us the hedge ratio, it gives us a participation rate, et cetera.

What this means is that per call, we should trade half a unit in the underlying, and that means because we have a hundred calls that's 50 units in the underlying that we need to trade. And then the next question of course will be, okay, do I have to buy or do I have to sell these 50 units i.e. let's say stocks for example.

Well, as explained on the previous slide, a long call gives you long Delta, which basically means you have a long exposure to the asset.

So the Delta hedge that you now have to do in order to be neutral or in order to neutralize the existing long exposure that will have to be sell 50 stocks.

Right? Now, let's assume this is done, okay, so we bought this a hundred calls with Delta of 0.5.

Now we sold 50 stocks.

What does that do when a spot price or when the spot price now let's say starts to move right after we've done that uh, transaction.

And let's think about a scenario where the, um, spot price of the asset decreases by $1.

So right after we've done these transactions, the spot price of the underlying decreases, so as per the option Delta that is given to us at 0.5, what we know is that an increase, uh, sorry, a decrease in the underlying price of $1 should lead to a loss of 50 cents per option.

It's a loss because we have long Delta, but the price goes down, okay? 50% participation or Delta of 0.5, that's how you work out the 50 cents per option.

Now remember, we are long a hundred options.

That means this $1 move in underlying spot price, that's gonna lead to a total loss of negative $50.

Okay? So that's what's happening.

The option position has lost $50 in value, there's nothing we can do about it.

However, we're not just long the options, but we're also short 50 stocks.

And this 50 stock short position is now gonna generate a positive p and l of $1 per share simply because the price of the underlying has decreased by $1.

And as we sold 50 shares, to be Delta hedged, which we could now theoretically buy at a $1 lower price, that means this hatching position apologies has now allowed us, or you know, has basically generated a gain of $50. We're ignoring transaction costs use, et cetera for for simplicity, right? And so if we take these two components together, then what we do see, indeed we are Delta hedge.

And this initial $1 price move had actually a total net impact on our portfolio of indeed zero um, dollars, right? So, this means now basically that there's a relatively simple technique to eliminate our Delta i.e. our directional exposure.

However, that of course requires to trade the underlying, at least this hedge that we suggested requires this.

And that's why we always have to consider the liquidity of the underlying when we're thinking about the appropriate size of an option position.

So if the option position here in this example would not have been a hundred calls, but instead a hundred thousand calls, then the initial Delta hedge would've been to sell 50 sell shares.

And the question we have to ask ourselves before entering into this position is, is this something that we expect to be able to do quickly without impacting the market? So it's really comparing the liquidity of the underlying market with the require Delta hedge to make sure that we're not having any unpleasant surprises when we're trying to execute our Delta hedge.

And now one final point on Delta hedging sometimes the counterparty of the option trade might be looking to Delta hedge as well, okay? So going back to our long call example, the call seller might want to be or might want a Delta hedge as well. And what they have to do is buy the underlying to hedge their short Delta position.

And in such a case, it really could be beneficial for both parties to agree on a Delta exchange.

So instead of us both now executing the Delta hedge in the market, us buying and, and them selling or the other way around, we can basically agree to address these associated or to to address this Delta hedge just directly in a transaction or a direct transaction between the two involved counterparties. So I buy from you, you sell to me, and then, we are both having achieved our Delta hedge without, you know, executing the hedge in the market with all the associated risks like market slippage and the sorry, market impact slippage, et cetera, et cetera. So basically the counterparties can decide to provide the Delta hedge directly, to each other.

Now, of course, make no mistake if there's any reporting requirements, they still have to be met.

But you know, as a general concept, I hope you can see how this makes transactions potentially low risk and also more efficient, right? So let's hammer this point, on Delta home because I think Delta is really important to understand.

And what I'd like you to do is briefly think about the three questions here, uh, on this slide just to make sure we're feeling super comfortable with the interpretation of Delta and thinking about the hedges, et cetera, et cetera.

So scenario is an investor has bought 1001 months, put options on an asset that trades at $55 in the spot market.

Now the delta in absolute terms is given at a 0.46.

Now the first question that I would like to you to think about for a couple of seconds is, is the investor long or short delta? I give you 10, 15 seconds to think about it, then we're gonna debrief, right? So what's the situation here? The investor is long a put, and as we said earlier, a long put position gives you a short Delta position and this basically means that the Delta here in this particular case is negative.

So it should really say negative with 0.46 or negative 46%.

That brings us to the second question.

How much would you expect the option premium to change if the underlying price rolls by $1? Again, I give you 10 seconds to think about and then we're gonna quickly go through this.

Okay? So Delta was, described by us as a negative number of 0.46, and that consequently means that we should expect the premium of the option to decline by 46 cents.

In case of a $1 price increase.

Price goes up negative Delta means the value or the option premium goes down and it should go down by 46% of the actual price increase.

And that is then 46 cents.

We have a total position of 1000 shares.

So the total loss would then be here, $460 on that position.

And then finally leads us to question number three, what would be the Delta hedge for this position from the investor's perspective? Now, I don't think we have to go through this or in an amount of a great amount of detail because we already said the main components where long 1000 put options, the Delta is negative 0.46.

So basically we are short 460, um, shares on that, or at least synthetically, right? We're not effectively short, but you know, the exposure is equal to being short 40, well 460 shares and that's then basically what the investor needs to buy, buy 460 shares.

And you can easily crosscheck that, uh, with the answer in question two.

So what if you bought 460 shares and the underlying price went up by $1? Well, you're losing four 60 dollars on the option position, but you're gaining $460 in your Delta hedge.

And so these numbers all make perfect sense. Good.

Now hopefully this was quite intuitive and now what we want to do next is have a look at how Delta actually behaves.

Okay? And so we're taking a particular option type of option here.

It's a European call option.

The strike is a hundred dollars, we're assuming it's a non-dividend paying asset.

We're also assuming zero interest rates.

So we're kind of keeping, really all this background noise out as much as we can.

Now what we're gonna uh, investigate here is, you know, how does Delta off an option change? We've been talking about the fact that Delta sits somewhere between zero and one.

In absolute terms, at least for vanilla options, that should be the case.

But what actually influences if Delta is more towards zero or more towards one? Well, conceptually I'd like to think this is quite intuitive, right? Because if Delta can be seen as a likeness factor or participation participation ratio, as we said, a higher Delta basically means the option behaves more than the underlying or more similar to the underlying right? And this should really be the case. If you think about it, the higher the probability of the option is to be exercised.

So if an option, for example, was deeply in the money, and well, let's look at this, dark blue line here for starters, right? So what is this? This is a, the Delta of an option that has one day to expire across different asset price scenarios. So let's say we're looking at a scenario where the asset price currently is at 115, the strike of the call we own is a hundred and there's only one day left to expire. Now of course it depends a little bit on the volatility of the asset, but you know, let's leave this aside for now.

It feels like this option is almost, if you know, if this is not a crazy volatile asset, at least this option is almost guaranteed to be exercised tomorrow, right? And as a result, this option should really behave very much like the underlying yes, legally speaking, this might still be an option.

Contract not, not, might be this still is an option contract. So it says in the contract, we have the right to buy the asset tomorrow at a hundred, but if the asset price is at 115, this almost, you know, practically becomes an obligation, right? So, there's no uncertainty whether or not the option will be exercised.

And uh, that means the option should start to behave more like a forward, right? Because that's effectively what is, we know we're gonna exercise this option tomorrow.

What we don't know is the exact amount of money that we're gonna get in exchange, because that depends on the price of the underlying tomorrow.

But it's reasonably safe to assume that we're gonna exercise because, you know, we're $15 in the money.

And assuming that there's no crazy volatility behind this that makes the option almost guaranteed, uh, to be exercised.

And you might have heard of Fords being referred to as Delta one product.

And that's because the value of a Ford contract moves more or less perfect, perfectly in line with a change in the underlying price. And if you feel like 115 is maybe a little bit too close, then you change the example to 150 or to 200 Eden.

And that kind of, when you realize, okay, that Delta ops the option should really be close to one.

It's never gonna be exactly one, but it's, you know, it's very, very close to one simply because we're almost guaranteed to exercise this.

Now that's the in the money example.

Now on the other hand, if an option is deeply out of the money, so we're looking now at this part here of the chart.

So let's say we have this asset trading at 85, we have a strike of 100 and the option expires tomorrow.

Reasonable or you know, average volatility sort of assumed here that is not really an option contract anymore either.

That's basically a loss of premium waiting to be booked, right? Because there's hardly any chance for this option to be exercised.

Even if the asset would rally from 85 to maybe 86, 87, 88 decent percentage move, that would still not really change the possibility or the probability of the option to be exercise. And so that shouldn't really influence how much like the underlying the option behave.

So for those scenarios, Delta will be reasonably close to zero.

Again, if 85 is too close to you, then you just sort of go and say, what if the asset would trade at $10? Well, you know, it should really be the option should really move by almost nothing, even if the price doubles from 10 to 20, right? So that means really, really deep out of the money options have to tend to have Deltas close to zero really deep in the money options.

We'll have a Delta closer to one, but that also means as we're sort of going through the moneys space.

So here we're starting with apologies starting with OTM, here is at the money and then we're going towards in the money.

What happens is we're starting in the out of the money space with a Delta of close to zero.

Then we're going towards at the money and at the money options, at least that of visual impression we can get, tend to have a Delta of around zero.

0.5.

Now, important to note that this is not quite true in reality, but I think it's a sort of rule of thumb that we're going to use going forward in this session.

Because we don't have to price options and calculate Deltas, but we want to make sure we have the conceptual understanding and then I think it's quite useful to use that Delta 0.5 for add the money as some sort of a, a guidepost.

So as the call that we're looking at here now goes from out of the money to add the money and then goes what's in the money, the option Delta increases and that means that the option premium increasingly behaves more like the underlying itself as we said.

Now this is true for all options that we're showing up because we have only focused on one and that was the one day to expire. But you can see here there's also the Deltas of a one month to expire, two months to expire is three months to expire.

Now they all show a very similar behavior in the sense that, you know, as the option goes in the money, the Delta goes up.

But what we can also see is that shorter dated options, especially the one with a, you know, one day to expire, tend to have at least in certain ranges or in certain region a much more rapidly changing Delta.

So when we're looking at just this this area here, right, the Delta of the very short dated option changes much more rapidly than for the one months, two months, three months option.

And again, I think that's quite intuitive and especially if you think of extreme examples.

Again, so let's say we are looking at two call options.

Both have a strike of 100 and the spot price is 100 as well.

One of the two calls expires tomorrow and the other one expires in 10 years from now.

Now at the moment, both options are at the money.

Because if you look at the slide header, and I told you before, we are assuming zero interest rates and no dividends here, which basically means spot equals forward.

And therefore both options will have a delta of approximately 0.5 or 50%.

Now if now the stock moves up to 110 or 115, again, this will, as we've seen, have a significant impact on the delta of the one day option, as now this option is much more likely to be exercised tomorrow and therefore we should really start to see a change in the behavior of the option premium because it should behave almost perfectly like the the the asset price right Now. How does that then differ for the 10 year option? Well, yes, even for the 10 year option, the likelihood that this option will be exercised might have increased slightly, but realistically how much has really changed? Yes, the option is in the money, but then there are 10 years left to expiry.

So I would argue that the probability that this option will be exercised in a decade from now doesn't, or has not really changed in a meaningful way. And therefore it's hopefully understandable that the option won't suddenly start to behave a lot more, uh, like the underlying i.e. this same move in underlying price leads to very different changes in delta and that is driven by a couple of factors.

But the one we've discussed now here, over the last couple of minutes is clearly the time to expire.

Okay, um, now, um, we're actually with this understanding of Delta, gonna have a look at the second order Greek that I have already mentioned earlier, and that's called gamma.

Now, as we've seen on the previous slide, the Delta of an option is not stable, but it changes as, for example, the underlying price changes.

There are other factors influencing that, but let's focus on this one here.

And this fact that Delta actually changes has quite important practical implications because as we said, Delta gives us the hedge ratio and the changing Delta then means it changing hedge ratio.

And what that finally means is that hedges we've put in place must be adjusted if we want to remain Delta hatched as the underlying moves.

And now we've talked about how important Delta and Delta hedging is, earlier.

And so I think it's quite understandable then that option traders really need to have information about how stable their option Delta is and how stable was that their hedge ratio is.

And that's the information that Gamma gives us.

Now, I said earlier, Gamma is a second order degree, which means it doesn't measure the impact of something on the option premium, but instead it measures the impact of a change in the underlying price on the options Delta.

So it's basically the delta of Delta or the rate of change of Delta and therefore measures, the steepness of the Delta function at any particular point.

So what that means is that if an option has a high gamma, it has relatively rapidly changing Delta values, which of course is information really worth having from a delta hedging perspective. Because if you have a relative fast changing Delta, this might require more frequent reeding then in case you have an option that does have a Delta that's pretty stable in comparison.

Now the one question that comes up then in this context of course is, am I long or short gamma when I'm having a certain option position.

Now, unlike for Delta, the answer to this question doesn't depend on whether we are looking at a call or a put, but instead on whether we are long or short, the option.

Now any long option position, regardless if this is through puts or calls, will lead to a long gamma position.

And what that means is that Delta will always change in your favor as the market moves, it will increase as the underlying price goes up and it decreases as the underlying price goes down.

So let's walk through this, for a long call position, right? So long call position means positive gamma to start with.

Now let's look at what happens to the delta of our long call as the spot increases.

And we can go back, to the previous slide just to, visual or remember the visual.

So let's say we're starting with an at the money, strike here.

So the spot is at a hundred and now the spot increases to, I don't know, let's make it up 104.

And that for the one day option has already led to a pretty meaningful increase in the options, Delta, right? So that means we have now a higher participation of the option in a scenario where the underlying price moves in your favor, i.e. it moves up.

That's the case for a long call.

How does things compare for long put? Well, that means that short delta position that we have by buying a put gets smaller and that means larger in absolute terms, which again means higher participation.

When prices move in the right way, i.e. they are going down.

So remember, regardless of your buying a call or a put, you have a long gamma position. That means Delta moves in favor and for short positions of both calls and puts isn't just the other way around.

And short positions generally mean you have a short gamma.

Of course, one of the key takeaways of all this i of the existence of gamma is that Delta changes when the underlying changes.

And that means the Delta hedging, if you decide to use it, needs to be dynamic.

I there's no way to just buy or sell an option, hedge the Delta and then basically walk away and come back until expiry because you're perfectly hatched.

That doesn't work unless of course you're using other options, to hatch.

But if you're just using this, transactions in in the underlying, then this will not be enough.

You need to have a dynamic, approach to fix.

And then of course, it makes sense to think a little bit about how Gamma generally behaves, right? To get a sense for which type of options have high gamma values, which kind of options tend to have relatively low gamma. And as we said on the previous slide, options with a high gamma are those that have a rapidly changing Delta.

And that of course brings us back to, okay, whenever Delta is rapidly changing, that's options that should have a relative high Gamma.

And that basically was, if you remember our Delta discussion, the case when options are, relative short or have relative short time left until expiry, and then also, they have relative or strikes that are relatively close to at the money.

So basically if we're looking at this slide here, this one is the gamma of our one day option for the various different scenarios of the asset price.

Remember the strike is a hundred. We have a one day option which gives us a right to buy the asset at a hundred.

And that underlying traits at a hundred, that's the definition of an add the money option.

And that's also the definition of a short time to expiry option simply because there's only one day left to expiry. And since no surprise here that this option has a very, very significant, Gamma value.

If you move away from the at the money strike and you're going deep in or deep out of the money, you see that gamma rapidly declines and that basically reflects that Delta becomes, more stable.

And then also if you extend the maturity range 1, 2, 3 months, you can see again that Gamma declines quite significantly and that's just due to the longer time to expire. So short dated at the money options, that's where gamma is, significant.

Okay, so that's all nice to know, you might argue, but what's a practical relevance, right? Well the practical relevance is, as we already sort of alluded to, is that through the existence of gamma, uh, option hedging or Delta hedging an option becomes somewhat challenging, right? because you know, and this of course as I said, we're we're hedging them back to back with buying the same option that we sold and and et cetera. But you know, that's very often not done for obvious streets.

Let's use this example here and and think about it because it illustrates quite well where the challenge is, right? So assume, we are a market maker now and we sold those thousand puts that we've looked at earlier from the perspective of an investor, right? And now in that scenario, we wanna think about four questions here.

So what we've actually given to you on the slide as well is a Delta table.

So here at the bottom you can see the Delta values of those options that will apply to certain, underlying prices.

So as we're starting, this case study, we've been given that we sold this a thousand put options, we have a spot price of 55 or an underlying price of 55, and that means the Delta of our put option is 0.46. That's no surprise, it's because it was 0.46 in the previous case using the exact, same numbers.

Now, first question is of course, you know, what is the Delta of the option? How many shares do we have to buy or sell in order to be Delta neutral? And that is of course just the opposite of the question we looked at earlier, right? Just now from the perspective of the market maker, we sold thousand puts, which means we're having a long Delta position now of 0.46.

So we can just write that as positive or plus 0.46, and that is the hedge ratio per option if you wish.

So that means we're having basically expressed in units of underlying a position of 460 shares, on the long side and to Delta hedges, we now need to sell 460 shares.

And let's say we're doing this.

And that's if you remember the exact opposite of what the investor needed to do, as we said, now, let's say we've done that, the hedge in plays, and then immediately after that the stock price rallies 2 57.

So now what is changing is that our option, the put option, uh, has basically moved out of the money and then as a consequence, Delta in absolute terms has decreased and it's now standing at 0.39 according to the table.

And we're ignoring Volatil. We're assuming volatility hasn't changed, et cetera, et cetera. We're just looking at a change of, the underlying spread privacy, right? So now with the Delta of 0.39, what this means is the correct hedge at that point would be to be short 390 shares, but we are short 460 shares from the initial Delta hedge.

So basically we are right now over hatched and to become Delta neutral again, we would have to buy 70 shares back at the current market price of 57.

So now you start to see the problem we sold shares at 55, which we're now buying back at 57.

And now let's go to question three and um, and, and explore the next scenario.

And that is that, you know, basically a few moments later after we adjusted our Delta hedge, the price of the underlying goes down again 2 55.

And with that Delta is basically back to where we started at 0.46, which means now we are under hedge, we're only short 390 shares because we bought some back, remember? But according to the new Delta, which is basically also the old Delta, we should be short 460 shares.

Now how to fix this, well, easy selling additional 70 shares, but really what we've done now in this round trip is we have bought 70 shares at 57 and then sold them at 55.

And hopefully this is straightforward to see that this results in a loss of, uh, 140 US dollars.

Now, before you now start thinking, okay, option selling is financial suicide, it's important to consider that in this context that, you know, we sold the option, which means we have been paid the premium.

So these $140 are not a net loss right now, but it's something that we should deduct from our option premium because, we have realized a loss there, but technically speaking, you know, you can sell an option as long as you don't lose more money Delta hedging in through other factors than the premium you, you charge for that upfront, then there's no loss position for you at all anyway.

So question four, right? How would the P&L have looked if you decided not to adjust the Delta hedge after the initial move up to 57? And what would've been the risk in not doing so? Now of course this is answering something with a benefit of hindsight, right? So we can say now,that we know how prices have moved, that it would've been better to do nothing at all at 57 because that would've meant if then prices dropped back to 55.

We did not experience this loss of 140 bucks, right? But of course the risk of not adjusting the Delta hatch at that point of 57 would've been that this wasn't just a move that, you know, would've been neutralized basically a couple of moments later, but that there's actually the beginning of a new trend and then price of the underlying would've continued to go up.

And then of course, you are in a short position in an environment where the market prices are rising and that would of course have led to potentially quite substantially higher losses than adjusting your Delta hedge.

So of course the right answer will always be known exposed, but that doesn't really make help us at that point when we have to make a decision, right? Now let's quickly look at this slide here because we've so far looked at the gamma position as a potential source of losses from the hedges perspective.

And remember, we sold the option here, so we were short gamma and that led to those $140 that we have described.

But of course, you can also look at Gamma and say, well there's, you know, there's opportunity in there.

And you know, that's something that's a strategy we might want to apply.

We might want to take advantage of this behavior of options that we just discussed, for example, when we expect that the market price of an underlying is gonna move a lot, but we're unsure about the direction. So that brings us into the world of volatility trading, by the way, right? So that's the, let's assume that's the market view we have.

What could we do? Well, the go-to solution to that could be to buy a straddle, which means we buy a call and a put, um, with the same strike.

And the Deltas of both options will mostly cancel each other out.

And if you want to be perfectly Delta neutral, you could calculate any residual Delta and then hatch that away as well.

So you have a long position in a call, a long position to put, and let's say you are perfectly Delta neutral.

So now what happens if the underlying price starts to move? What happens is that one of the two options will move in the money.

So let's say the price of the underlying goes up, what happens, the call that you own goes in the money that leads to an increase in Delta at the same time the put that you've bought goes out of the money, which means it decreases Delta in a absolute way.

And that basically means that those two moves taken together lead to a net long Delta position in an environment where market prices are rising.

So that will move in your favor.

And if it was the other way around, the price starts going down, then the put goes in the money, the call goes out of the money again, you would put pushed automatically in a short position in an environment where market prices are falling.

Now that of course sounds too good to be true, whereas the catch well to, you know, go into this long position of two options in the first place, you need to pay premium twice.

So you really run the risk that there's not a sufficient amount of volatility over the life of these two options that, you know, and, and you won't be able to recover the total premium outlay you had completely.

And that you will be faced a financial loss at the end.

But, let's leave Gamma and Delta behind for now and quickly introduce the two other option Greeks that we wanted to talk to or talk about here.

Theta and Vega. And we're starting with Theta and Theta measures the impact of the passage of time on the option pre.

Now all L's equal right option premium for calls and pers should decline as time goes by and the option move closer to expiry as simply the time value of these options decline.

Right now a long option position, regardless of put or call therefore comes with a negative status, i.e. all else equal.

The option premium will be lower tomorrow than it is today, which from a buyer's perspective of course is negative because if you think about it, of the context of a mark to market valuation buying an option today at $1 and tomorrow or Monday, it's only worth 98 cents that clearly has negative percussions for the, for the P&L.

And when you look at Theta at the sort of like charts that display how theta behaves, then I think you sort of like think that looks vaguely familiar, right? Because it just looks like the flip side of Gamma. And so a good question to ask you in this context is if this is actually a incidence or if these two i.e. Gamma and theta are actually somehow connected, and in my view the link between those two, value i gamma and feas actually, you know, quite intuitive because gamma can be understood as some sort of like option specific risk factor, right? Because unlike a forward position, which could, you know, which, which risk can really easily be understood through the Delta and this Delta can be hatched and we can effectively walk away and come back when the food expires.

If we're ignoring other things here for now, the option Gamma introduces really this additional layoff complexity.

And we have discussed that just in the previous Delta hedge right now.

On the other hand we have Theta which represents the time decay of an options price.

And from a buyer's perspective, of course that can be seen as a cost of holding an option.

But if you look at this from the perspective of the option seller, you can almost interpret this as the compensation you are receiving as the option seller for taking on the risk that's associated with the options.

Gamma i.e. was the risk that delta does change, right? And then there's this most or one of the most fundamental concepts in finance that says risk and reward for taking this risk should be sly related.

So it does from that angle makes perfect sense to see that a high absolute theta value is usually found for those options which have high Gamma.

And that brings us to add the money and short time to expire.

So once again, the same picture here and that then brings us ladies and gentlemen to the last Greek that we wanted to recap here today.

And that's Vega.

Now Vega measures the sensitivity of the premium to a change in volatility.

Now more specifically it looks at the implied volatility here, but let's leave this topic for another day and just sort of understand as a general, volatility concept.

Now, as we intuitively discussed on the first slide, the premium of a call and a put is positively correlated with volatility.

So if volatility of the underlying increases, then the uncertainty regarding the option exercise increases and consequently the time value of the option increases.

And with it the premium of the option increases all else being equal, right? So what we can say from that is that buyers of options, both calls and puts are exposed to Vega. They are long Vega, they benefit from an increase in a volatility.

But how do different options then compare regarding their Vega i.e. Which type of options tend to be more sensitive to a change in volatility and which type of options tend to be less sensitive? Well, I said earlier a change in volatility affects the time value of the option, not so much the intrinsic value.

So from that, what one could sort of derive is that options which have a higher proportion of the total premium being given by time value.

They should have a higher Vega, they should respond to a change in volatility, more sensitively than options where the time value component of the total is, is only just a small faction of the total, premium.

Right? So that then means to answer which options will have a high Vega. We have to think about which options have relatively high time value and time value is relatively high for at the money options we talked about that already is sort of all about the uncertainty of exercise, which is greatest at, at the money strike if you wish, and also increase waste time to expire. And that's exactly what the chart here in front of us shows for short dated options.

The Vega is highest when the strike is at the money and viga increases with time to expire.

Now, just to I think clarify one thing which sometimes might lead to some sort of confusion, if, while we talked about Gamma, we basically indicated that the, I don't wanna say problem, but, but gamma, or the gamma trade that we have that we have described briefly is some sort of a volatility trait and now we're seeing Vega, which is the sensitivity to volatility.

So where now exactly is the difference between Gamma and Vega or what does, you know, really gamma trading versus volatility or volatility trading by taking a Vega position now? So we talked about it in the context of gamma.

What gamma does is it really gives us exposure to actual realized price movements, right? Where we say, okay, I buy this underlying and, sorry, I buy the call and the put, and if the underlying goes up, I rehash my delta. If it goes down, I rehash my Delta.

And so I need prices of the underlying to move, so that I'm able to extract a positive P&L out of this trade. When you switch to Vega, then things change, quite significantly in the sense that of course there's no complete disconnect between the actual realized volatility of underlying the and the implied volatility.

But remember I said that's measuring the sensitivity to implied wall, which sort of is almost like an indicator that shows how much demand there is for an option.

And so that's, there is a difference which we hopefully will be able to discuss at a different day because now we've reached the end of this session.

I thank you so much for your participation today.

Hope you found it beneficial.

I look forward to hopefully, welcoming you back on one of those sessions in the near future.

Have a great weekend and thank you very much for now. Bye-Bye.